Yes, they would. The pressure to advance western society was driven by exploitation of the frontiers resources and since the battles fought on the frontier tended to be small affairs with limited manpower, the potential for recruitment and retaliation post 1865 was quite substantial.

Of course news of terrible defeats affect public opinion adversely, but they can also create outrage, and consider that the sense of threat against christian white communities from 'out of control savages' would have meant a president demanding stiff response to that threat, real or imagined, and as we know - conquerors are often highly regarded in human socities - something american politicians would have been well aware of.

Really makes you wonder if the natives ever had a chance against the European.

Very likely, the native american communities were doomed by force of population alone, putting aside other factors, such as technology, etc.

It was closer than what is assumed today. The rare early victories of the European settlers were enough to give them some momentum, which allowed more people to settle and therefore more people to spread disease and so on so forth. Initially the tech difference wasn't that decisive and by the 1800's the natives often had modern guns, for example at the Little Bighorn the Sioux and Cheyenne warriors not only had more guns than Custer's detachment, but faster-firing ones as well. And an independent Maya country came very close to being formed as a permanent entity in 1848, and survived as late as the 1900's. If some of the first European victories went the other way, which is easier than you might thing, then the settlers would lack the same drive and momentum that got them the hemisphere so quickly, at least if you consider 400 years to be quick.

For example, as far as the Maya example mentioned goes, if a Yucateco town fell to the Maya rebels just a few days earlier, the Maya would've gotten to Merida that much faster and have been able to take it before the harvest season, therefore winning the entire Yucatan and making Guatemala feel very uneasy with a new Maya-run Native-American state north of them that has British backing, as it did in real life. Guatemala might then find itself falling to Maya rebels. The North American natives might have troubles creating states so late, but the Iroquois Confederacy likewise had British support and was a relatively stable entity only falling because of the Americans. It's possible they might have survived if all the tribes rather than just a couple supported the fledgling American rebellion, or if the British won the Revolutionary War instead. So yeah, not too hard to find ways for at least some Native-Americans to survive as independent nations. If you went even further back in time it could be even easier by getting rid of Cortes. The butterfly effect from even that would be tremendous, even if people like to believe the conquest was inevitable.

As far as best leader goes, I voted Crazy Horse, but Rigoberta Menchu is a good choice as well. I want to also put out Benito Juarez if he counts, he was Mexico's best president to be sure.

Mexico is in North America. So is Guatemala by most definitions. If Mexico is not North American then Geronimo isn't eligible as he was born in Mexican territory. Unless for some reason the modern borders of Mexico are what define geography.

In any case, if I could I'd amend my vote to either Hiawatha or Deganawida, but neither seem to be available choices. Shame.

Chief Joseph, a man of peace and dignity shamefully treated by the US authorities. The Nez Perce War was his attempt to retreat with his tribe towards Canada. Constantly he thwarted the US army set against him even though hampered by women, children and the aged.and never left behind one wounded person.
The petty vindictiveness is well illustrated in Sherman sending four hundred men, women and children four hundred miles in unheated rail cars and held in a prisoner of war camp for eight months.

In at least one of his biographies, and in some contemporary descriptions, Crazy Horse is depicted as being unusual in appearance for a man of his people (i.e., having lighter hair and eyes, etc.) Does anyone have any theories about this?